Mark the tick against the correct answer in the following:
Let $\mathrm{Q}^{+}$be the set of all positive rationals. Then, the operation $*$ on $\mathrm{Q}^{+}$defined by $\mathrm{a} * \mathrm{~b}=\frac{\mathrm{ab}}{2}$ for $\mathrm{all} \mathrm{a}, \mathrm{b} \in \mathrm{Q}^{+}$
is
A. commutative but not associative
B. associative but not commutative
C. neither commutative nor associative
D. both commutative and associative
According to the question,
$\mathrm{Q}=\{$ Positive rationals $\}$
$\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a} * \mathrm{~b}=\mathrm{ab} / 2\}$
Formula
$*$ is commutative if $a^{*} b=b^{*} a$
$*$ is associative if $(\mathrm{a} * \mathrm{~b}) * \mathrm{c}=\mathrm{a}^{*}\left(\mathrm{~b}^{*} \mathrm{c}\right)$
Check for commutative
Consider,$a * b=a b / 2$
And,$b * a=b a / 2$
Both equations are the same and will always true .
Therefore , * is commutative ……. (1)
Check for associative
Consider, $(\mathrm{a} * \mathrm{~b}) * \mathrm{c}=(\mathrm{ab} / 2) * \mathrm{c}=\frac{\frac{a b}{2} \times c}{2}=\mathrm{abc} / 4$
And,$a^{*}\left(b^{*} c\right)=a^{*}(b c / 2)=\frac{a \times \frac{b c}{2}}{2}=a b c / 4$
Both the equation are the same and therefore will always be true.
Therefore, $*$ is associative $\ldots \ldots$ (2)
Now, according to the equations (1), (2)
Correct option will be (D)
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,